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Tag Archives: primes
Primality Tests III
SolovayStrassen Test This is an enhancement of the Euler test. Be forewarned that it is in fact weaker than the RabinMiller test so it may not be of much practical interest. Nevertheless, it’s included here for completeness. Recall that to … Continue reading
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Tagged cryptography, elementary, jacobi symbol, legendre symbol, number theory, primality tests, primes, programming
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Primality Tests II
In this article, we discuss some ways of improving the basic Fermat test. Recall that for Fermat test, to test if n is prime, one picks a base a < n and checks if We also saw that this method would utterly fail … Continue reading
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Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
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Primality Tests I
Description of Problem The main problem we wish to discuss is as follows. Question. Given n, how do we determine if it is prime? Prime numbers have opened up huge avenues in theoretical research – the renowned Riemann Hypothesis, for … Continue reading
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Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
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Topics in Commutative Rings: Unique Factorisation (3)
Example 1: The Gaussian Integers Z[i] Let’s pick the norm function N : Z[i]{0} → N where N(a+bi) = (a+bi)(a–bi) = a2+b2. We know that N is a multiplicative function, i.e. N(r)N(s) = N(rs). Instead of checking this by brute force, we write N(x) = x·xc, where (a+bi)c = abi is the conjugate of a+bi. It’s easy to … Continue reading
Topics in Commutative Rings: Unique Factorisation (2)
In the previous article, we imposed certain finiteness conditions on the ring (specifically a.c.c. on principal ideals: that every increasing sequence of principal ideals is eventually constant), then proved that unique factorisation holds if and only if all irreducible elements … Continue reading
Posted in Notes
Tagged commutative rings, euclidean domains, irreducibles, prime ideals, primes, principal ideal domains, ring theory, rings, UFDs, unique factorisation
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Topics in Commutative Rings: Unique Factorisation (1)
Unique Factorisation: Basics Throughout this post, let R be an integral domain; recall that this means R is a commutative ring such that whenever ab=0, either a=0 or b=0. The simplest example of an integral domain is Z, the ring of integers. What’s of interest to … Continue reading
Posted in Notes
Tagged commutative rings, irreducibles, prime ideals, primes, ring theory, rings, UFDs, unique factorisation
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